Look at your front derailleur (left hand) as "groupings". On flat(ish) surfaces, you should be able to start in the middle (2) of a triple. I have never ridden a double, up front, so can't say. So, for me on my triple front, 9 speed rear:
Spend most of my time in the middle front (2). Start in 6, as that is my "easy, but not too easy" flat surface move out gear. Quickly shift to 7 for flat running. If hitting the downhills, shift to 3 (large chain) up front and gradually to 8, then 9, as the downhill steepens. For gradual uphills, I downshift to 2 in front, then 6-5-4-3 in rear as needed. I might even shift down to 2, as experience (and advice here) have proven that the chain and derailleurs can handle it.
Coming up stonger hills is mildly more involved: My general tendency is to drop two gears, in the rear (making pedalling easier) to keep momentum (though one at a time is perfectly fine). Depending on how these old legs feel, I might not hit the granny (small) in front. As others have said, keep your eyes on traffic AND the lay of the land: much easier to start shifting BEFORE you start to grind... saves the knees and muscles
So, for my commute: (front derailleur/rear derailleur) 2/6 from dead start, a few strokes to speed and 2/7. Down the lane to short downhill (no shift needed) approaching steep, short, uphill. Quick sequence, approaching uphill: 2/-6-5-4, then down to the granny (1) up front, and 1/3-2-1 as I hit the steep part. Cresting onto a gradually uphill parking lot, 1/2-3-4 and 1/3-2-1 as it steepens, another crest and 1/2-3-4, front shift 2/4-5-6 to flat running again.
First substantial hill (noting that real hill warriors would laugh): 2/6-7 starting down, upshift front 3/7-8-9 achieving max speed for the upslope, and 3/7-5 downshift front 2/5-4-3 as necessary to make the top. Most everything else is variations of the above. My daily commute is just 5 miles/8 km one way and does involve rolling hills at stretches.
Some of the fantastic people here understand gears far better than I, and major thanks to the one that got my understanding by putting the math into a feet/revolution formula: ((front teeth/rear teeth x tire size in inches) x 3.14) /12
So, for my current setup: 1/2/3 front gears of 22/32/42 gear teeth; and 1-2-3-4-5-6-7-8-9 rear gears of 34-30-26-23-20-17-15-13-11 gear teeth.
From hardest to easiest pedalling
3/9 26ft
3/8 22ft
3/7 19ft
3/6 17ft
3/5 14 ft
2/7 14.5ft
2/6 13ft
2/5 11ft
2/4 9.5ft
2/3 8.5ft
2/2 7.3ft
1/4 6.5ft
1/3 5.75ft
1/2 5ft
1/1 4.4ft
So, you can see where the shifting usually will occur, in the overlaps. On the ground, I tend to shift the front (up or down) when the rear is in 7th gear, or 4-3. So a normal progression for me would be 1/1-2-3-4 SHIFT 2/4-5-6-7 SHIFT 3/7-8-9. Noting that when I had larger rings up front (changed within the last year), I was shifting about one rear gear easier to overcome the bigger front.
A lot to chew on, perhaps. Hopefully it is presented to make sense. If not, there are no dumb questions... fire away!